Witnesses at an upcoming trial over no-hire agreements in Silicon Valley should not be allowed to offer evidence that Apple co-founder Steve Jobs was “a bully,” four major tech companies argued in a court filing.

Tech workers filed a class action lawsuit against Apple, Google, Intel and Adobe Systems in 2011, alleging they conspired to avoid competing for each other’s employees in order to avert a salary war. Trial is scheduled to begin at the end of May on behalf of roughly 60,000 workers in the class, and defendants say damages could exceed $9 billion.

The case, which is closely watched in Silicon Valley, is largely built on e-mails among top executives, including late Apple CEO Steve Jobs and former Google CEO Eric Schmidt.

For instance, after a Google recruiter solicited an Apple employee, Schmidt told Jobs that the recruiter would be fired, court documents show. Jobs then forwarded Schmidt’s note to a top Apple human resources executive with a smiley face.

In a joint court filing late last week, the companies told District Judge Lucy Koh in San Jose, Calif., that they were not seeking to bar Jobs’ interactions with other witnesses about the no-hire agreements. However, opinions based on other evidence, like Walter Isaacson’s best-selling biography about Jobs, should be kept out of trial, they said.

An attorney for the plaintiffs could not immediately be reached for comment on Tuesday, nor could a representative for Apple. Representatives for Google, Intel and Adobe declined to comment.

“Plaintiffs’ only purpose for offering this testimony would be improper — to cast Mr. Jobs in a bad light,” the companies said in the filing.